√951 at a glance
- Exact value
- √951
- Decimal (10 places)
- 30.8382878902
- Rounded
- 30.8 · 30.84 · 30.838
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.838288
- Prime factorization
- 3 × 317
- Cube root
- 9.833924
How to simplify √951
The prime factorization of 951 is 3 × 317. Every prime appears only once, so there is no pair to bring outside the radical — √951 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 951, 3 and 317 appear an odd number of times, so √951 is irrational and 30.8382878902 is a rounded value.
Where √951 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √951 lies between 30 and 31. 951 is 51 above 900 and 10 below 961, so the root is closer to 31.
- Straight line between 900 and 961: 30.8361 (0.01% low)
- Tangent from 30, i.e. 30 + 51 ÷ 60: 30.8500 (0.04% high)
- Tangent from 31, i.e. 31 − 10 ÷ 62: 30.8387 (0% high)
For √951 the tangent at 31 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 951 is just 10 below 961.
Finding √951 with the Babylonian method
If a guess is too big, 951 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√951) in one step.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 951 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 30.6774193548 | 30.8387096774 | 3 |
| 2 | 30.8387096774 | 30.8378661088 | 30.8382878931 | 8 |
| 3 | 30.8382878931 | 30.8382878873 | 30.8382878902 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √951 = 30.8382878902 to every decimal shown.
√951 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √951 the pattern is [30; 1, 5, 5, 2, 3, 1, 1, 1, 9, 1, 1, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √951 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 30/1 | 30.0000000000 | 8.4 × 10⁻¹ |
| 31/1 | 31.0000000000 | 1.6 × 10⁻¹ |
| 185/6 | 30.8333333333 | 5.0 × 10⁻³ |
| 956/31 | 30.8387096774 | 4.2 × 10⁻⁴ |
| 2,097/68 | 30.8382352941 | 5.3 × 10⁻⁵ |
| 7,247/235 | 30.8382978723 | 1.0 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 951y² = 1. Its smallest solution in positive whole numbers is x = 224,208,076, y = 7,270,445.
√951 in geometry and everyday measurements
- 951 square feet is 88.4 m². Laid out as a square — a small house footprint or a lot — it is about 30.84 ft (30 ft 10 in) on a side.
- 951 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √951 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √951 as its space diagonal.
Square roots near √951 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √948 | 2√237 | 30.7896 | No |
| √949 | √949 | 30.8058 | No |
| √950 | 5√38 | 30.8221 | No |
| √951 | √951 | 30.8383 | No |
| √952 | 2√238 | 30.8545 | No |
| √953 | √953 | 30.8707 | No |
| √954 | 3√106 | 30.8869 | No |
- The cube root of 951 is about 9.833924.
- Squaring undoes the root: (√951)² = 951, while 951² = 904,401 — the number whose square root is 951.
Frequently asked questions
What is the square root of 951?
The square root of 951 is √951, about 30.8382878902. The negative root, −30.838288, also squares to 951.
Is the square root of 951 rational or irrational?
Irrational. 951 is not a perfect square — it falls between 900 and 961 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √951 be simplified?
No. 951 = 3 × 317 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √951 rounded to two decimal places?
√951 ≈ 30.84 to two decimal places (30.8 to one, 30.838 to three). Check: 30.84² = 951.1056, close to 951.